Exponential EquationsExplanation, Solving, and Examples
In arithmetic, an exponential equation arises when the variable appears in the exponential function. This can be a scary topic for children, but with a bit of direction and practice, exponential equations can be determited easily.
This blog post will discuss the explanation of exponential equations, kinds of exponential equations, proceduce to solve exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The primary step to solving an exponential equation is determining when you have one.
Definition
Exponential equations are equations that consist of the variable in an exponent. For example, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two primary items to keep in mind for when trying to determine if an equation is exponential:
1. The variable is in an exponent (signifying it is raised to a power)
2. There is no other term that has the variable in it (besides the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The primary thing you must notice is that the variable, x, is in an exponent. Thereafter thing you should not is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
One more time, the primary thing you should notice is that the variable, x, is an exponent. The second thing you must note is that there are no more terms that have the variable in them. This means that this equation IS exponential.
You will come upon exponential equations when you try solving various calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are essential in math and play a pivotal responsibility in working out many computational questions. Thus, it is important to completely understand what exponential equations are and how they can be utilized as you go ahead in arithmetic.
Types of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly easy to find in everyday life. There are three major kinds of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the easiest to solve, as we can easily set the two equations equal to each other and work out for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be created the same using properties of the exponents. We will put a few examples below, but by changing the bases the same, you can observe the same steps as the first event.
3) Equations with distinct bases on both sides that cannot be made the same. These are the trickiest to figure out, but it’s possible through the property of the product rule. By increasing two or more factors to identical power, we can multiply the factors on both side and raise them.
Once we are done, we can set the two latest equations equal to each other and figure out the unknown variable. This blog does not contain logarithm solutions, but we will tell you where to get assistance at the very last of this article.
How to Solve Exponential Equations
After going through the definition and kinds of exponential equations, we can now understand how to work on any equation by following these simple steps.
Steps for Solving Exponential Equations
We have three steps that we need to ensue to work on exponential equations.
Primarily, we must recognize the base and exponent variables within the equation.
Second, we are required to rewrite an exponential equation, so all terms have a common base. Subsequently, we can solve them utilizing standard algebraic rules.
Lastly, we have to solve for the unknown variable. Since we have figured out the variable, we can plug this value back into our initial equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's take a loot at a few examples to note how these steps work in practicality.
First, we will solve the following example:
7y + 1 = 73y
We can see that both bases are identical. Thus, all you need to do is to rewrite the exponents and figure them out using algebra:
y+1=3y
y=½
So, we change the value of y in the respective equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a further complex sum. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a common base. Despite that, both sides are powers of two. By itself, the solution comprises of decomposing both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we work on this expression to conclude the final answer:
28=22x-10
Perform algebra to solve for x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can recheck our workings by replacing 9 for x in the original equation.
256=49−5=44
Keep seeking for examples and problems over the internet, and if you use the laws of exponents, you will turn into a master of these theorems, solving most exponential equations with no issue at all.
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